Question

select the correct answer from each drop-down menu.
in the figure below, $u \parallel v$
prove: $\angle 3$ and $\angle 5$ are congruent.

statements reasons

1. $u \parallel v$ given
2. $\angle 1 \cong \angle 7$
3. $\angle 5 \cong \angle 7$ vertical angles theorem
4. $\angle 1 \cong \angle 3$ vertical angles theorem
5. $\angle 3 \cong \angle 5$

Explanation:

Step1: Identify reason for ∠1≅∠7

Since \(u \parallel v\), ∠1 and ∠7 are alternate exterior angles, which are congruent by the alternate exterior angles theorem.

Step2: Identify reason for ∠3≅∠5

We know ∠1≅∠7, ∠5≅∠7, and ∠1≅∠3. By transitive property of congruence (if \(a≅b\) and \(b≅c\), then \(a≅c\)), we can conclude ∠3≅∠5.

Answer:

1. For the reason of ∠1 ≅ ∠7: alternate exterior angles theorem
2. For the reason of ∠3 ≅ ∠5: transitive property of congruence